THE MAY MEETING IN FRESNO

The four hundred eighty-first meeting of the American Mathe­matical Society was held a t Fresno State College, Fresno, California, on Saturday, May 3, 1952. Approximately 75 persons attended, in­cluding the following 59 members of the Society:

H. L. Alder, H. M. Bacon, W. G. Bade, A. K. Bell, L. D. Berkovitz, E. W. Beth, F. C. Biesele, W. W. Bledsoe, L. M. Blumenthal, F. H. Brownell, J. G. van der Corput, P. H. Daus, Douglas Derry, D. J. Ewy, F. D. Faulkner, G. E. Forsythe, N. S. Free, R. M. Hayes, M. R. Hestenes, P. G. Hodge, P. J. Kelly, L. D. Kovach, Cornelius Lanczos, B. J. Lockhart, Charles Loewner, M. W. Maxfield, L. F. Meyers, E. D. Miller, F. R. Morris, T. S. Motzkin, Ivan Niven, L. J. Paige, T. K. Pan, R. S. Phillips, D. H. Potts, W. T. Puckett, F. M. Pulliam, C. H. Rawlins, Jr., R. M. Redheffer, J. B. Robinson, R. M. Robinson, Leo Sario, M. M. Schiffer, I. J. Schoenberg, Abraham Seidenberg, R. G. Self ridge, M. A. Shader, Max Shiffman, W. H. Simons, T. H. Southard, M. L. Stein, Robert Steinberg, C. B. Tompkins, J. L. Ullman, W. R. Wasow, M. A. Weber, Frantisek Wolf, H. H. Wolfenden, F. H. Young.

By invitation of the Committee to Select Hour Speakers for Far Western Sectional Meetings, Professor I. J . Schoenberg, of the Institute for Numerical Analysis, National Bureau of Stand­ards, Los Angeles, delivered an address entitled On smoothing operations and related topics. Professor Schoenberg was introduced by Professor Ivan Niven. There were sessions for contributed papers in the morning and afternoon, presided over by Professors F. R. Morris, H. M. Bacon, and P. H. Daus. Following the meeting there was a guided tour of Pine Flat Dam, now under construction.

Following are abstracts of papers presented at the meeting. Papers with abstract numbers followed by "t" were presented by title. Paper number 490 was presented by Professor Steinberg, number 498 by Professor Kelly, and number 502 by Dr. Forsythe. Dr. Reid was in­troduced by Dr. J. W. Odle and Mr. Bennett by Professor F. C. Biesele.

ALGEBRA AND THEORY OF NUMBERS

473/. D. R. Morrison: Boolean product rings.

A Boolean product R o B is defined for every ring R and every Boolean ring B, differing from the tensor product in that the left distributive law, r o (&1+&2) —r o fa +f2&, holds only if &i&2 = 0 or 2r = 0. Every nonzero element of JR O B is uniquely expressible in the form X)*-i r< ° &*» where r , ^ r t ^ 0 , fo^O, fabj — 0 for j^i. Where they are meaningful, the commutative and associative laws and the distributive laws with respect to weak direct sums apply to the Boolean product operation, and GF(2) is a unit with respect to the operation. The idempotent Boolean ring (R o B)°

493

494 AMERICAN MATHEMATICAL SOCIETY Uuly

(see Bull. Amer. Math. Soc. Abstract 58-1-52) is isomorphic to R° o B. If R is bi-regular, so is R o B and relations are determined between the ideals of R, B, and Ro B. For any Boolean ring B, and any prime, P, GF(p) o B is a £~ring. Conversely, for any £-ring R, R° is a Boolean ring and R is isomorphic to GF(p) o R°. This estab­lishes a correspondence between Boolean rings and p-rings, which is biunique up to an isomorphism. (Received March 13, 1952.)

474. T. S. Motzkin: The multi-index transportation problem.

Let Xaw'ctn* a*~l» * ' ' » a* (Ö»>1, ^ = 1, • • • , »), be unknowns. For a given r, r —1, • • • , « —1, make all choices k — (\^ii< • • • <in-r£n; ottV'",otin_^). It is shown that the extreme solutions of the system X)*«i«i• "anssaC*» *ai«2-**«n^0 (*ne

summation running over all at with «7^1, • • • , in-r) lie, for every given system of constants c&, in the ring generated by the Ck, if and only if r = 1 and no more than two a% exceed 2. (Received March 13, 1952.)

4752. R. M. Redheffer: Power series and algebraic numbers. Suppose ƒ(2) = ^2,anz

n is rational, is regular at 2=0, has algebraic coefficients, and is not a polynomial. Let m be the multiplicity of that root in the denominator of ƒ which has maximum multiplicity. Then (apart from a set of at most m values of z) the value of the function g{z)~ YLan^n/n\ is transcendental when the value of z is algebraic. (Received March 17, 1952.)

476. Abraham Seidenberg: Some basic theorems in differential algebra {characteristic p, arbitrary).

The theorem of the primitive element, the chain theorem, and the Hubert Nullstel-lensatz, all appropriately formulated, are established for differential algebra of arbitrary characteristic. Simplified proofs of the known results in the case of character­istic 0 are included. The theory of transcendency and the theory of separability are also considered. (Received March 13, 1952.)

ANALYSIS

477/. R. E. Bellman: Some mock hydrodynamical equations. Pre­liminary report.

The equation (1) ut-{-uux ~IAUXX was used by J. M. Burgers, Proc. Acad. Sci. Amsterdam vol. 43 (1940), as a mathematical model to illustrate the theory of turbulence. It was shown by E. Hopf, Comm. Pure and Applied Math, vol, 3 (1950), that this equation could be linearized by means of the change of variable, w= — 2fx(log 4>)z, <t> satisfying the equation ufax—fa* It is shown in the present paper that « = — 2ju(log <£)», Ü = — 2/x(log 4>)y furnishes a solution to the two-dimensional analogue of (1), ut+uux-{"VUy~ix(uXx+uyy), vt+uvx+Wy~n(vxx-\-vvy), if 0 satisfies the equation /j.(<t>xx+<t>yv) —4>t, and similarly for three dimensions. These explicit solu­tions may now be used, following the one-dimensional treatment of Hopf, to de­termine rigorously the behavior of the solutions as /*—»0. If the pressure terms are included in the hydrodynamical equations, the equation for 4> is fx(<f>xx+<j>yy) +£0 — 0*. (Received March 17, 1952.)

478. F. H. Brownell: Vector analysis on real separable Hubert space. Preliminary report.

i952] THE MAY MEETING IN FRESNO 495

Consider complex-valued functions \f/(f) over /Gfe, the square summable real se­quences and the prototype of real separable Hubert space. Following Hille, for suit­able functions \f/ one can define the notion of a vector derivative W(f), where for * real V^(f)G/2 with | |w| |-1 |^(/+«)-^(f)-(V^(f), w)|-»0 as |M|-*0, * £ / * This can be interpreted in terms of the /2 coordinate derivatives, and using basis coordinates the generalized Laplacian V2 can be defined invariant under change of basis. In order to integrate on k and still have an abelian group, it seems k must be reduced to some square summable infinite torus Xt X~{xÇzk\ -*hn<xnZ*hn each n) with Z^°°(W2

< -|~ oo required, a modification of Jessen's torus integration. This gives a locally compact abelian topological group, which thus possesses Fourier transforms with inverses, and one can verify the expected relationship of Fourier transforms to partial derivatives with respect to k coordinates, as well as to the generalized Laplacian V2. (Received March 13, 1952.)

479/. R. S. Phillips: On the generation of semigroups of linear operators.

Let T(t) be a semi-group of linear bounded transformations on a Banach space £ to itself, strongly continuous on [0, oo) with r (0 )= I . Set co(£)~log | |^(ö | | and define <S(o>) to be the Banach algebra of set functions a where ||a|| = / ^ exp [«(£)] I da\ and the product is given by convolution. Let wo=inf co(£)/£ and set <f>(\, a) =ƒ* exp (X£)da for Re (X) ^w0. Assume further that limô-0sup^o [w(Ö—«(£+$)] < oo. Then a necessary and sufficient condition that ot(t, • ) be a semi-group of mono­tone nondecreasing functions in <S(w) (with <f>[\, a(t, • ) ]—>1 as /—»0 uniformly in every bounded subset of Re (X) ̂ w0) is that t"1 log #[x, a(t, '^^tnK+J^iexp [(X—w0)£] — \)d\p-\-a where w^O, a is real, and ^(£) is monotone nondecreasing on (0, oo) satis­fying the conditions fl%d\p<<x> and ƒ" exp [a)(£)—co<£]d\l/< <*. For a semi-group a(t, ' ) of this description S(t) =f*T(£)da(t, £) is again a semi-group of linear bounded transformations on £ to itself, strongly convergent to / a t £ = 0. If A is the infinitesimal generator of T{t) and B that of S(t), then for x in the domain of A, Bx = mAx -\-fo [exp (--woÖTXÖff—x\d4/-\-ax. In addition if X0 belongs to the spectrum of A, then /"^[Xo, <*(/, • )] belongs to the spectrum of B. (Received December 18, 1951.)

480. R. S. Phillips: Perturbation theory for semi-groups of linear operators.

Those properties of a semi-group of linear bounded operators strongly continuous to the identity at the origin which persist under a linear bounded perturbation of the infinitesimal generator are called stable. It is shown that the property of being a semi-group or a group is itself stable. If A generates T($) and A+B (with B bounded) generates S(s), then S($) — Z^-o ^«W w n e r e So(s) — T(s) and Sn(s) =f'QT(s—(r)BSn-i((r)dor. If T(s) is uniformly continuous for s>0, then so is S(s). However the property of being eventually uniformly continuous is not stable. If T(z) is holomorphic in a sector and bounded near the origin in every sub-sector, then the same is true of S(z). If the resolvent of A is completely continuous, then so is the resolvent of A -\-B. It is possible to develop a "finite" elementary divisor theory for A when the resolvent of A is completely continuous. If A(s) ~A+B(s) where A is the infinitesimal generator of a semi-group of operators and B(s) is a strongly con­tinuously differentiate family of bounded linear operators, then there exists a unique strongly-continuous one-parameter family of bounded linear operators U(s) such tha.t dU(s)x/ds~A(s)U(s)x for all xÇ:D(A) and such that J7(0)=*I. (Received Febru­ary 22, 1952.)

496 AMERICAN MATHEMATICAL SOCIETY Uuly

481/. R. M. Redheffer: A basis theorem for partial differential equa­tions.

Let X = (x, y, z, • • • , w) denote the independent variables, £7= U(X) the de­pendent variable, and V=(U, Uz, • • • ) a finite-dimensional vector with £/and some of its partial derivatives as components. A partial differential equation is an equation of form ƒ (X, V) = 0. If ƒ is continuous and if a linearly closed family of solutions Ua

solves ƒ(X, V) — 0, then there is a finite system of linear homogeneous partial dif­ferential equations A • F^O, B • V=0, • • • , such that (1) each Ua solves the system and (2) each solution of the system solves ƒ (X, V) =0. The result follows from simple properties of finite-dimensional vector spaces. (Received March 17, 1952.)

482/, R. M. Redheffer: Differential equations and the Huygens principle.

Let U(x, y) ^(l/(2ir)1fi)/2OQeity[A(t)]xF(t)dtf where A is measurable, A^O, A ^ oo ; and let U satisfy one and the same partial differential equation for all F giving suffi­ciently strong convergence of the integral. Then log A(t) is equal almost everywhere to an algebraic function of /, and the functions U all are solutions of one and the same linear homogeneous partial differential equation with constant coefficients. (Received March 17, 1952.)

483. R. M. Redheffer: Operators and the Huygens principle. Let <t>ab be a class of operators which transfer a disturbance ƒ(#, y) from x—a to

x~b:f(b, y) — <j>abf(a, y). These give an abstract formulation of Huygens' principle in one dimension. Assuming linearity, a closure condition <f>ac — <l>ab<l>bc, and boundedness in two senses, representation theorems are given. The simplest result (for "uniform media") is 4>ab=iT~lté,~aT, where T is the Fourier transform in the L2 sense, T~l the inverse transform in the L sense, and A(w)£i> for all p. (Received March 17, 1952.)

484. W. P. Reid: On the relaxation method of solving differential equations.

Allen and Severn (Quarterly Journal of Mechanics and Applied Mathematics vol. 4, Part 2 (1951) p. 209) have stated that "It is, in fact, only to problems of jury type that the relaxation method is directly applicable," where jury type means equal num­ber of boundary conditions at each of two points. This statement is incorrect. The relaxation method may be applied directly to solve differential equations of any order with the n boundary conditions distributed in any way among P points (1 ^kP^n). Their method of raising the order of the differential equations to make them of jury type is a method of fooling oneself and results primarily in extra work. These remarks, by the way, apply also to problems governed by partial differential equations. Also, the applications of the relaxation method have apparently been re­stricted so far to bounded regions. This is not necessary. The region may be un­bounded, and a condition may or may not be specified at infinity. (Received March 10, 1952.)

485/. Leo Sario: An extremal method on arbitrary Riemann surfaces. I.

A method is indicated for constructing a function which minimizes a prescribed expression mina given class F of functions defined on an arbitrary Riemann surface

1952] THE MAY MEETING IN FRESNO 497

R. Consider the class {P} of analytic functions with the development P=p+ip = l / 3 + 5 ^ i r av%v m a fixed parameter disc K\\z\ ^ 1 , and with a single-valued real part on R. Write a = Re (#i) and denote by |3 the (ideal) boundary of R. Let X be a real parameter, — 1 ^ X ^ l . Then (1) there is in {P} a unique principal function P x s = ^ x + ^ x = = l / 2 + S ^ a\vz

v which minimizes the expression m\=X-27ra-f-J'ppdp. (2) This minimum is mI(^x) = (ir/2)[(l+X)2ai-(l-X)2a_i] where aX = Re (a\i). (3) mx(p)-mx(px)='D(p-px), the Dirichlet integral. (4) Px = (l+X)Pi/2-Hl-X)P_i/2. (5) a i ^ a ^ a _ i for functions p with fppdp^O. Definition: the P-span <rp = a_i--ai. (6)<rp = min VMfppdp^VMfepodpo. (7) l/<rP = min (l/27r)P(/) = (l/27r)P(/o) where (ƒ) is the class of single-valued harmonic functions ƒ on P with the development ƒ+*?= ZXz v in j£, Re ( k ) « l , and / O + ^ O - C P - I - P O A P . (8) The class {P} de­generates to a single function if and only if the P-span vanishes. These results remain valid for the subclass IQ] of {P}, defined by the restriction fdp=0 along each di­viding cycle on P, if the P-span is replaced by the corresponding Q-span. For planar surfaces, IQ} consists of single-valued analytic functions and the Q-span coincides with Schiffer's span. Earlier results concerning the span appear as special cases. (Received March 10, 1952.)

486/. Leo Sario: An extremal method on arbitrary Riemann surfaces. II.

The boundedness of p-i—px and q~i—q\ (Q-i—Qi for planar surfaces) is used to establish a unified approach to classification problems on Riemann surfaces. The following abbreviations are employed: H, harmonic single-valued nonconstant; Kt

harmonic single-valued nonconstant, such that the conjugate function has no periods along dividing cycles; A, analytic single-valued nonconstant; W, meromorphic nonconstant; B, bounded; D, with a finite Dirichlet integral; E, omitting a set of values with positive area. By combinations of letters, classes of functions are denoted with the corresponding combinations of properties. Let OF be the class of Riemann surfaces not admitting functions belonging to a given class F. Denote by Sp and SQ the classes of surfaces with vanishing P-span or Ç-span, respectively. Then (1) OHBCOHBD = OBD = SP> (2) OKBCOKBD = OKD = SQ, (3) OWE = OABCOAD. Earlier results, concerning H and A, are included in these relations. The last equality (1) states that there are functions HD on R if and only if the P-span vanishes. The rela­tions (2), applied to planar surface, express existence properties of single-valued analytic functions. (Received March 10, 1952.)

487/. Leo Sario: Construction of functions with prescribed properties on Riemann surfaces.

Let G be the complement P—D of a compact subdomain D of an open Riemann surface R, D being bounded by a finite set a of analytic Jordan curves. The (ideal) boundary of R is denoted by /3. Let v be a single-valued function, harmonic on «, let X be a real parameter, — 1 ^ X ^ l . Denote by {«} the class of single-valued harmonic functions in G with u~v on «, fadfl—0. There exists in {u} a unique function u\ which minimizes the expression fpudü+\faudü. The function u\ is associated with » b y a normal linear operator L\. This theorem remains valid for the subclass {u°} of {u} defined by the restriction fdu — 0 along every dividing cycle. Both u\ and u^ are constructed by the extremal method reported in the above abstracts. The linear operator method (Trans. Amer. Math. Soc. March 1952) leads to the following theorem. Let s be a single-valued real function on Dt harmonic on a. The condition

498 AMERICAN MATHEMATICAL SOCIETY [July

fads~0 is necessary and sufficient for the existence of a function p on R such that (1) p—s is harmonic on D, (2) fpudü+xfaudü is minimized by p among all harmonic functions u in G with u~p on a and fdu~0 along a (or else along every dividing cycle). (Received March 10, 1952.)

488. Leo Sario: Capacity of the boundary and of a boundary com­ponent, I.

Let h(z) be a harmonic function in \z\ ^ 1 with h(0) =0. Write 5 = log \z\ -\-h(z). Denote the curves \z\ ~e~x

t \z\ « 1 by a and b, respectively. Lemma: mint $—min0 5 ^ 1 . On an arbitrary open Riemann surface R with (ideal) boundary 0, let {S} be the class of analytic functions with the development 5=s-H$ = log z-\- 2JIT avZv

in a parameter disc \z\ ^ 1 , and with a single-valued real part s. There is in {S} a unique principal function Sp such that min fpsds—fpspdsp. Definition: the capacity of j8 is C/3=e""*0 where kp —(l/2w)fpspdsp. For planar domains, C/j coincides with the conventional (logarithmic) capacity. Consider the class { U] of analytic functions with a single-valued modulus on R and with the development £/= X£lT WX&i — l) in | s | 5£1. In { U], there is a unique principal function Up=esP such that min D(J7) ~D(Up)>=*irCI . Suppose now /3 is analytic. For every analytic function w with single-valued modulus on R such that \w\ ^ C " \ the relations |w>| < | Z7/j|, H>'(0)<1 hold unless w^eiaUp. Denote by M the class of nonconstant analytic functions with single-valued modulus on an arbitrary Riemann surface R. The subclasses of func­tions with bounded modulus or with finite Dirichlet integral are denoted by MB and MD, respectively. There are functions MB or MD on R if and only if C$>0. (Received March 10, 1952.)

489/. Leo Sario : Capacity of the boundary and of a boundary com­ponent. II.

Let 7 be a fixed boundary component of an arbitrary Riemann surface R with boundary 0. Consider the subclass {T] of {S} defined by fds=*2ir along every cycle separating y from 0 = 0, and fds~0 along every other cycle. There exists in {T} a unique principal function Ty such that min fptdt^fptydiy. Definition: the capacity of a boundary component y is Cy = e~ky with ky = (l/2ir)fptyd"ty. Let { V) be the subclass of {U\ defined by fd arg u=*2ir along cycles separating y from 2=0, fd arg w=0 along other cycles. On a planar R, there is in { V} a unique principal function Vy

=eTy such that min Z>(F)=Z>(F7)==7rC~2. There exist on R univalent bounded or univalent Dirichlet bounded nonconstant functions if and only if the capacity of at least one boundary component is positive. (Received March 10, 1952.)

490. Robert Steinberg and R. M. Redheffer: Simultaneous trigo­nometric approximation.

Given real functions f(x) and g(x) in L2, how small can Ia*f\\f(x)+ig(x) ~~ 2/Lo at)einx\ 2dx be made by a suitable choice of the real a£hs, possibly subject to the constraint £/Lo a£)2^M? Expressions for the minimizing a™'s and the mini­mum I are obtained. In the unconstrained case, J can be made arbitrarily small as n—>oo only for conjugate f unctions ƒ (#) and g(x) if 6 = 7r, but for arbitrary functions if 0 <b O . The latter case leads to matric relations analogous to the ordinary Parseval identity. The following limitations on the approximation are obtained: (1) In general, there is no series approximation (a™ independent of n) for which I-»0. If there is a series approximation, it is unique. (2) The following classes of function pairs ƒ, g

1952] THE MAY MEETING IN FRESNO 499

are equivalent: (a) Those which admit a polynomial approximation with ƒ—»0 and S/Lo afcW2 —M. (b) Those which admit a series approximation with I—>0 and ]C*Loa* ^Af. Hence, in general, the approximation in (a) is not possible. (Received March 17, 1952.)

491. J. L. Ullman: Hankel determinants injunction theory. Pre­liminary report.

Consider the power series aoz^+aiz"2-^ • • • —/(s), and let A(n, k) be the Hankel determinant of order k with an+i+j-& as the element in the ith row and jth column. To each sequence of integers [kn\ associate the number [&»]' = lim sup»-* \A(n, kn)\

p(-n)t

p(n) = l/(kn(n+kn — l)). Hadamard proposed the problem of finding constructive means of determining the singularities of an analytic function from the coefficients of a single Taylor series expansion. In this direction he showed (J. Math. Pures Appl. (4) vol. 8, p. 101) that when the elements of [kn] are equal, [kn]

f is related to certain poles of ƒ(z). Pólya (Math. Ann. vol. 99, p. 687) considers the case in which Hm».*» kn/(n+kn) =€>0, and shows that the [kn]' are related to the entire set of singularities of ƒ(2). The large gap in these results suggested that [kn]

f for the case lim„̂ oo kn— °°, limn-» kn/(n+kn) =0, in which the rate of growth of kn is intermediate to the other cases, would be related to intermediate properties of ƒ (s). A precise rela­tionship is established for f unctions ƒ (z) —JldiA(t)/(t—z), p,(t) real and nondecreasing on (0, 1). Of independent interest is a generalization of the well known fact that if .4(0, k) is ultimately zero, then 11(f) is a step function; namely, it is shown that if limjuoo 1-4(0, k)\1/k2=0, then tx(f) has zero derivative on (0, 1), except for a set of measure zero. (Received April 15, 1952.)

492. W. R. Wasow: Asymptotic solution of the differential equation of hydrodynamic stability in a domain containing a transition point.

The differential equation « ( 4 ) +L/_i a/(*)«<*^>+X,X)Li &b(*)«(|-*)a-0, with &o(0) =0, bd (0) ?^0, ô2(0) 5*0, is studied, in a complex neighborhood 5 of # = 0, for large values of the complex parameter X. The a,(#) and bh(x) are supposed to be analytic in S. This problem is important in the study of the onset of turbulence in viscous flows. In a paper in Ann. of Math. vol. 49 (1948) pp. 852-871, the author constructed a fundamental system with known asymptotic behaviour in a ring-shaped domain surrounding the origin. It is now shown—independently of that paper—that the differential equation above possesses a fundamental system which, for large X, is approximated, everywhere in S, by functions that can be simply and explicitly ex­pressed in terms of solutions of the differential equation y(i)+\2(xy"-\-y) =0. The asymptotic properties of this latter differential equation are known from another paper by the author (Ann. of Math. vol. 52 (1950) pp. 351-361). Thus it is possible to give a complete asymptotic analysis in S of the more general differential equation, at least as far as the asymptotically leading terms are concerned. (Received March 3, 1952.)

APPLIED MATHEMATICS

493. P. G. Hodge:. Upper and lower bounds on the yield load of a square slab with a centered circular cutout. Preliminary report.

If an elastic-plastic body is loaded with a set of loads P, the collapse load cor­responding to P is defined as the smallest loads iiP, where ft is a scalar, under which the body will exhibit unrestricted plastic flow. The present paper is concerned with a

500 AMERICAN MATHEMATICAL SOCIETY [July

thin square slab with a centered circular cutout, subjected to uniform tensile stresses on each pair of opposite sides. The cutout factor X is defined as the largest multiplier X such that if (Tx, Ty) represents any collapse load for the square without cutout, the square with a cutout will not collapse under any load less than (\TX, xr t f). Upper and lower bounds are then found for the cutout factor. The analysis is based on the as­sumptions of plane stress and Tresca's yield condition. (The results presented in this paper were obtained in the course of research conducted under Contract N7onr-35810 between the Office of Naval Research and Brown University.) (Received March 10, 1952.)

494. Cornelius Lanczos : The smallest root of the characteristic equa­tion.

In vibration problems the absolutely smallest eigenvalues of the generally non-symmetric and nonreal matrix A are of specific interest. Since the usual iterative procedures suppress the small eigenvalues in favor of the large ones, a preliminary inversion of A is customarily demanded. The present procedure analyzes the problem in the orthogonal reference system of the Hermitian matrix C^A^A. The basic equation is written in the form Cx—\A*x~0. The smallest eigenvector xo of C is considered as the zeroth approximation of x. In first approximation X—XQ-\-\XI where xi is the solution of a linear set of equations. The associated quadratic equation gives a (generally complex) X which is close to the absolutely smallest eigenvalue of A and which can be further refined. This method, if applied to ordinary algebraic equa­tions, yields a simple and convenient algorithm for approximating the absolutely smallest root of an algebraic equation. (Received March 4, 1952.)

495*. R. M. Redheffer: Antennas with obstructions. If T, T"1 are direct and inverse Fourier transform operators and A(u)

=exp i(k2—u2)ll2t fe = 27r/X, then ATaTtT~1lLa operating on the original far pattern

F(u) gives the perturbed far pattern F0(u) due to an obstruction dXx—a character­ized by the transmission function t{y). Let an arbitrary obstacle be characterized by T(u, v), the transmitted plane wave in "direction" v — {2ir sin 0)/X given a unit in­cident plane wave in "direction" u< Then (2w)1f2Fo(v)=A~a/AaT(uf v)F(u)du, and hence t(y) exists if and only if T(u, v)=<j>(u—v), for some 4>. A dense set in t, ƒ, or F yields a dense set in FQ, whence are obtained conditions of pattern readability and methods of computation. The perturbed pattern is approximable as a finite sum of expressions TAa/(u — &), the transform being taken as a Sommerfeld contour integral. Another approach gives modified Bessel functions. For estimates of FQ we have Fo = F-{~A~aH, where H exp {—imu), given explicitly, is slowly varying for some m. (Received March 17, 1952.)

496/. R. M. Redheffer: Inhomogeneous dielectric media. Let R be the complex reflection of a slab in which the dielectric constant is a con­

tinuous function of one coordinate, e = e(x). If JZ = — i(l+R)/(l —R) and —kez^u'/u, then u satisfies {uf ft)' + (2ir/\)2u~0. Formulas are given f or ^(z) in terms of 3(0), and for z(x) for every termination from z(x) for one. The polar form z — reid satisfies rd0/dr~ta.n 6(er2 — l)/(er2+l) when e is real, whence is obtained a condition for readability of a prescribed z. Regarding the medium as limit of thin uniform slabs gives a physical interpretation to the method of "product integration." (Received March 17, 1952.)

1952] THE MAY MEETING IN FRESNO 501

GEOMETRY

497. L. M. Blumenthal: Boolean geometry. I. A Boolean metric space 43 is obtained by associating with each pair of elements

p, q of a set an element d(p, q) of a Boolean algebra as distance, so that d(p, q)—0 if and only if p = q, d(p, q)=d(qt p), and d(p, q)+d{q, r£)d(pt r) (+ , D denote Boolean addition and inclusion, in the wide sense, respectively). Boolean geometry studies the invariants of 43 under the group of congruences (i.e., distance-preserving mappings). Any Boolean algebra is a space 43 with d(p, q):sspqf+pfq (juxtaposition and ' denoting Boolean product and complementation, respectively) and this paper begins the systematic study of these Boolean geometries by developing linearity notions and the theory of metric segments. Defining these in a suitable manner, they are shown to possess many properties of segments in ordinary metric spaces. Thus, (1) the length of a segment Sa is the distance d(a, b) of its end elements, (2) if Sh

a

Scb are segments and b is metrically between a, c then their set-union is a segment S°a,

and (3) for m >4, any m-tuple of 43 is on some segment if and only if each triple of the w-tuple has that property. (Received February 26, 1952.)

498. P. J. Kelly and L. J. Paige: Symmetric perpendicularity in hilbert geometry.

Define a family F of curves as follows: Every curve C in F is a simple, closed, convex curve. If supporting lines at p and q on C meet at w, and £ is any secant through w, cutting CdXm and n, then supporting lines at m and n exist, intersecting on pXq. The authors prove that the family F consists of all ellipses and all nondegenerate triangles. This result is used to show that the hilbert geometries in which perpendicu­larity is symmetric are hyperbolic geometries. (Received February 18, 1952.)

499. T. K. Pan: Characterization of the Gaussian curvature of a surface.

Let Vn be a hypersurface immersed in a Euclidean space Sn+i. Let v be a vector-field in Vn. Define the mean curvature of v at a point P as the square root of the sum of the squares of its n — 1 principal curvatures at P (the author's paper on Normal curvature of a vector-field, to appear soon in Amer. J. Math.). It is found that the hyperspherical curvature at P of Vn is equal to: (1) the product of the extreme mean curvatures of vector-fields in Vn at P, (2) the square of the product of the extreme mean curvatures of asymptotic vector-fields in Vn at P, (3) the product of the mean curva­tures of the n vector-fields at P corresponding to the principal directions for the Ricci tensor Ry when Vn is minimal. New interpretations of the Gaussian curvature of a surface S in an ordinary space are obtained: (1) the Gaussian curvature of S at P is the product of the principal curvatures of any two distinct conjugate vector-fields in 5 at P, (2) the Gaussian curvature of S at P is the negative of the square of the magnitude of the Gaussian representation of a unit arc along an asymptotic line from P in S. A simple proof of the theorem of Enneper is derived. (Received March 21, 1952.)

LOGIC AND FOUNDATIONS

500. D. W. Bennett: A set of axioms f or the natural numbers. Let N be a set for which the following axioms are satisfied: I. There exists a

mapping ƒ of the set of all nonempty subsets of N onto N. II, For any nonempty sub-

502 AMERICAN MATHEMATICAL SOCIETY [July

set P in N, f(P)ÇzP. III. There exists a reversible mapping g of the set N onto the set of all elements of N except ƒ (N). IV. For any subset P of N such that ƒ (P) ^/(iV), g~lf{P)@cP. The axioms are seen to involve no concepts except the familiar idea of mapping one set onto another. They lead immediately after appropriate definitions of " 1 " and "successor" to Peano's postulates for the natural numbers, and con­versely. (Received March 12, 1952.)

501. E. W. Beth: Existence of complete models f or extensions of the first-order predicate calculus.

Let T be a complete extension of the first-order predicate calculus. Every set 5 of expressions A (x) containing a free variable x defines, for every model M of P, a sub­set M(S) of the set of all elements of M. Let S be finite; then, obviously, M (S) will be nonempty if and only if S is consistent with P. For an infinite set 5 which is con­sistent with P, the set M(S) will not, in general, be nonempty for every model M of P. But it follows from the theorem of Löwenheim-Skolem-Gödel-Malcev that there is a model Mo of P which is complete in the following sense: for any set 5, Mo(S) will be nonempty if and only if S is consistent with P. It follows that M0(S) is non­empty if and only if, for some model M of P, M(S) is nonempty. For k = 2, 3, • • • an analogous result can be proved for sets of expressions A (xh x2) • • • , #*) contain­ing k free variables. (Received March 12, 1952.)

STATISTICS AND PROBABILITY

502. G. E. Forsythe and J. W. Tukey : The extent of n random unit vectors.

Let Wi, ' ' • , Wn be random unit vectors which are independently and uniformly (with respect to n — 1 dimensional measure) distributed on the unit sphere in «-dimen­sional space. Let A„ be the extent of these vectors, i.e., the signed «-dimensional volume of the parallelotope built on them. If Wi has cartesian coordinates wu, then A» = det (wn). A short geometrical proof of the following is given: for n*z2 the ratio | An/An_i | is the product of n — 1 independent positive quantities, each of which has the same incomplete beta distribution </P(w)«r(«/2){r(«/2~l/2)r(l/2)}""1

• w(w~"8>/2(l — u)~lf2du; the ratios A„/An_i, • • • , A2/A1, Ai are independent; the odd mo­ments E(A2

nfc+1)=0; the even moments E(A^*)=«!(«+2)! • • • («+2fc-2)!{0! 2! • • »

(2k-2)\}-1{n(n+2) • • • (n+2k-2)}~n. It is shown that the moments can also be obtained from the moments for the generalized variance; see S. S. Wilks, Biometrika vol. 24 (1932) pp. 471-494, esp. pp. 476-477. The second moment E(A^) had been given by S. Krochmal [Leningrad. Universitet. Uchenye Zapiski vol. 83, Ser. Mat. Nauk 12 (1941) pp. 150-198; Mathematical Reviews vol. 7 (1946) p. 487]. (Received March 10, 1952).

TOPOLOGY

503/. Barrett O'Neill: Essential sets and fixed points. Let ƒ be a self-map of a compact polyhedron X. A subset A of X is defined to be

essential if every map in some neighborhood of ƒ in Xx has a fixed point in A. Criteria are established that a subset be essential and that a subset contain an essential fixed point, as defined by M. K. Fort. The main tool is a fixed point index for sets similar to that of Leray and others. An elementary proof is given of the existence of an index and further properties derived. In special cases the essentiality of A is related to the

I9$2] THE MAY MEETING IN FRESNO 503

homological properties offtA, and X. Maps of locally compact spaces are considered through their compactifications. Using Cech theory, the methods apply to nonpoly-hedral spaces. (Received April 24, 1952.)

504/. R. H. Sorgenfrey: Mappings of plane convex sets. It is an almost immediate consequence of a theorem of Borsuk that if C is a com­

pact convex set of width w in a euclidian space and ƒ is a continuous real-valued map­ping defined on C, then some inverse set under ƒ has diameter at least as great as w. It is shown that if C is planar, then some inverse set has a component of diameter at least as great asw>(31/2)/3 in general and, in case C is centrally symmetric, at least as great as w(3l,2)/2. Simple examples exist to show that these numbers cannot be im­proved. (Received March 12, 1952.)

W . T . PUCKETT, Acting Associate Secretary